When two conducting or insulating inclusions are closely located in a composite, the gradient of the solution to the conductivity problem may become arbitrarily large in the narrow region in between them as the distance between the inclusions tends to zero. This phenomenon is known as field concentration, a central topic in the theory of composite material. We study the field concentration between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type, where the potential is allowed to be discontinuous across the interfacial boundaries. The imperfect bonding interface condition holds significant practical relevance, as it takes into account the contact resistance due to roughness and possible debonding at the interface, and it also approximates the membrane structure in biological systems. We discover a new dichotomy for the field concentration depending on the bonding parameter of the interface. The results are surprisingly different from the perfect interface setting. Based on joint work with Hongjie Dong and Zhuolun Yang.
